To compare with physical mesons, quark masses need to be incorporated. Brodsky and de Téramond adopted a natural ansatz, \(k_\perp ^2/x(1-x) \rightarrow (k_\perp ^2+m_q^2)/x + (k_\perp ^2+m_{\bar{q}}^2)/(1-x)\) , known as the invariant mass ansatz (IMA) [1]}. The resulting pion wave function reads,
\(\psi _\pi (x, \vec{k}_\perp ) = N(4\pi /\kappa )\exp \big [-(\vec{k}_\perp ^2+m_{u,d}^2)/(2\kappa ^2x(1-x))\big ]\) ,
where \(N\)  is a normalization constant, \(m_{u,d} = (m_u + m_d)/2\)  is the light quark mass. The pion mass becomes nonzero,
\(M_\pi ^2 = \int \frac{{\mathrm {d}}x}{2x(1-x)} \frac{{\mathrm {d}}^2k_\perp }{(2\pi )^3} \big | \psi _\pi (x, \vec{k}_\perp ) \big |^2 \frac{m_q^2}{x(1-x)}.\) 
